Problems of Evolution; Banach Spaces

Abstract

The laws of classical physics are causal or deterministic, and that leads to the concept of a well-posed initial-value problem. Roughly speaking, a detailed knowledge of the state of a system at time t = t₀ enables one to predict the subsequent states for all t > t₀. This chapter and the next two are devoted to the study of such problems. Differential equations are usually involved, and one must decide what is physically acceptable as a solution of the equations, and what are the appropriate initial and auxiliary conditions. A physical principle that guides the proper formulation of the problems is that there should be exactly one solution for every initial state, and the solution should depend continuously on the initial state, in a sense to be explained. It will be seen that Banach spaces provide the appropriate abstract setting for these problems. Most of the discussion is for linear problems; non-linear ones, for which the theory is quite fragmentary, are discussed briefly in Chapter 17.